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Tentukan nilai x yang memenuhi persamaan logaritma berikut! a. logx+log(x+1)=log2

Pertanyaan

Tentukan nilai x yang memenuhi persamaan logaritma berikut!

a. logx+log(x+1)=log2      

Pembahasan Soal:

Ingat 

alogf(x)=alogbf(x)=b 

Perhatikan perhitungan berikut 

logx+log(x+1)log[x(x+1)]log(x2+x)x2+xx2+x2(x+2)(x1)x+2x========log2log2log22000ataux1=02x=1 

Uji syarat numerus 

Untuk x=2 

x2>>00tidakmemenuhi 

Untuk x=1 

x1x+11+12>>>>>00memenuhi000memenuhi 

Jadi, nilai x yang memenuhi persamaan tersebut adalah x=1.  

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 13 September 2021

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Pertanyaan yang serupa

Jika , tentukan nilai  yang memenuhi persamaan tersebut.

Pembahasan Soal:

Ingat sifat-sifat logaritma:

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript a space b minus log presuperscript a space c end cell equals cell log presuperscript a space b over c end cell row cell log presuperscript a space a to the power of n end cell equals n row cell fraction numerator 1 over denominator log presuperscript a space b end fraction end cell equals cell log presuperscript b space a end cell row cell log presuperscript a to the power of m end presuperscript space b to the power of n end cell equals cell n over m log presuperscript a space b end cell end table  

Ingat ada persamaan logaritma berlaku jika log presuperscript h open parentheses x close parentheses end presuperscript space f open parentheses x close parentheses equals log presuperscript h open parentheses x close parentheses end presuperscript space g open parentheses x close parentheses comma space a greater than 0 dan a not equal to 1 maka f open parentheses x close parentheses equals g open parentheses x close parentheses dengan syarat h open parentheses x close parentheses greater than 0 comma space h open parentheses x close parentheses not equal to 1f open parentheses x close parentheses greater than 0 dan g open parentheses x close parentheses greater than 0

Diketahui fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses minus log presuperscript 16 space open parentheses x squared minus 4 x plus 4 close parentheses end fraction equals negative 2 maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses minus log presuperscript 16 open parentheses x squared minus 4 x plus 4 close parentheses end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator x minus 2 over denominator x squared minus 4 x plus 4 end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator x minus 2 over denominator open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space begin display style fraction numerator 1 over denominator x minus 2 end fraction end style end fraction end cell equals cell negative 2 end cell row cell fraction numerator 1 over denominator log presuperscript 16 space open parentheses x minus 2 close parentheses to the power of negative 1 end exponent end fraction end cell equals cell negative 2 end cell row cell log presuperscript open parentheses x minus 2 close parentheses to the power of negative 1 end exponent end presuperscript space 16 end cell equals cell negative 2 end cell row cell negative log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 end cell equals cell negative 2 end cell row cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 space end cell equals 2 row cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space 16 end cell equals cell log presuperscript open parentheses x minus 2 close parentheses end presuperscript space open parentheses x minus 2 close parentheses squared end cell row 16 equals cell open parentheses x minus 2 close parentheses squared end cell row 16 equals cell x squared minus 4 x plus 4 end cell row cell x squared minus 4 x plus 4 minus 16 end cell equals 0 row cell x squared minus 4 x minus 12 end cell equals 0 row cell open parentheses x minus 6 close parentheses open parentheses x plus 2 close parentheses end cell equals 0 end table  

diperoleh nilai yang memenuhi x equals 6 atau x equals negative 2  

Syarat numerus 1)

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell greater than 0 row x greater than 2 end table   

syarat numerus 2)

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 4 x plus 4 end cell greater than 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses end cell greater than 0 row x greater than 2 end table  

Dari syarat numerus mengharuskan x greater than 2. Dengan demikian nilai x yang memenuhi adalah x equals 6.

 

Roboguru

Jika  dan  maka

Pembahasan Soal:

Ingat sifat-sifat pada bentuk logaritma yaitu 

  • scriptbase log invisible function application f left parenthesis x right parenthesis end scriptbase presuperscript a equals scriptbase log invisible function application b end scriptbase presuperscript a rightwards arrow f open parentheses x close parentheses equals b
  • scriptbase log invisible function application b to the power of m end scriptbase presuperscript a equals m times scriptbase log invisible function application b end scriptbase presuperscript a
  • scriptbase log invisible function application b end scriptbase presuperscript a plus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application b c end scriptbase presuperscript a

serta ingat juga logaritma dengan basis 10.

Sehingga akan diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 to the power of x end cell equals 400 row cell log invisible function application 5 to the power of x end cell equals cell log invisible function application 400 end cell row cell x log invisible function application 5 end cell equals cell log invisible function application open parentheses 4 times 100 close parentheses end cell row cell x log invisible function application 5 end cell equals cell log invisible function application 4 plus log invisible function application 100 end cell row cell x log invisible function application 5 end cell equals cell log invisible function application 2 squared plus log invisible function application 100 end cell row cell x log invisible function application 5 end cell equals cell 2 times log invisible function application 2 plus 2 end cell row x equals cell fraction numerator 2 times log invisible function application 2 plus 2 over denominator log invisible function application 5 end fraction end cell row blank equals cell fraction numerator 2 times 0 , 3010 plus 2 over denominator 0 , 6989 end fraction end cell row blank equals cell fraction numerator 0 , 6020 plus 2 over denominator 0 , 6989 end fraction end cell row blank equals cell fraction numerator 2 , 6020 over denominator 0 , 6989 end fraction end cell row blank equals cell 3 , 72 end cell end table

Jadi, dapat disimpulkan bahwa jika log invisible function application 2 equals 0 , 3010 dan 5 to the power of x equals 400 maka x equals 3 , 72.

Oleh karena itu, jawaban yang benar adalah B.

Roboguru

Jika  dan  maka ...

Pembahasan Soal:

Diketahui : 

log presuperscript 3 space x minus log presuperscript 3 space y cubed equals 3 log presuperscript 3 space x squared plus 4 space log presuperscript 3 space y equals 6 

Ingat kembali bahwa :

bullet space log presuperscript straight a space straight a to the power of straight m equals straight m bullet space log presuperscript straight a space straight b to the power of straight m equals straight m space log presuperscript straight a space straight b bullet space log presuperscript straight a space straight b plus log presuperscript straight a space straight c equals log presuperscript straight a space left parenthesis bc right parenthesis bullet space straight a to the power of straight m straight a to the power of straight n equals straight a to the power of straight m plus straight n end exponent bullet space log presuperscript straight a space straight f left parenthesis straight x right parenthesis equals log presuperscript straight a space straight b rightwards arrow straight f left parenthesis straight x right parenthesis equals straight b space space space space space dengan space straight a greater than 0 comma space straight a not equal to 1 comma space straight b greater than 0     

Cari nilai log presuperscript 3 space y pada persamaan pertama

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 3 space x minus log presuperscript 3 space y cubed end cell equals 3 row cell log presuperscript 3 space x minus 3 space log presuperscript 3 space y end cell equals 3 row cell fraction numerator log presuperscript 3 space x minus 3 space log presuperscript 3 space y over denominator 3 end fraction end cell equals cell 3 over 3 end cell row cell 1 third space log presuperscript 3 space x minus log presuperscript 3 space y end cell equals 1 row cell 1 third space log presuperscript 3 space x minus 1 end cell equals cell log presuperscript 3 space y end cell row cell log presuperscript 3 space y end cell equals cell 1 third space log presuperscript 3 space x minus 1 end cell end table   

Substitusikan nilai log presuperscript 3 space y ke persamaan kedua

table attributes columnalign right center left columnspacing 0px end attributes row cell log presuperscript 3 space x squared plus 4 space log presuperscript 3 space y end cell equals 6 row cell log presuperscript 3 space x squared plus 4 space left parenthesis 1 third space log presuperscript 3 space x minus 1 right parenthesis end cell equals 6 row cell log presuperscript 3 space x squared plus 4 over 3 space log presuperscript 3 space x minus 4 end cell equals 6 row cell log presuperscript 3 space x squared plus log presuperscript 3 space x to the power of 4 over 3 end exponent end cell equals cell 6 plus 4 end cell row cell log presuperscript 3 left parenthesis x squared times x to the power of 4 over 3 end exponent right parenthesis end cell equals 10 row cell log presuperscript 3 space x to the power of 6 over 3 plus 4 over 3 end exponent end cell equals 10 row cell log presuperscript 3 space x to the power of 10 over 3 end exponent end cell equals 10 row cell 10 over 3 space log presuperscript 3 space x end cell equals 10 row cell log presuperscript 3 space x end cell equals cell fraction numerator 10 over denominator begin display style 10 over 3 end style end fraction end cell row cell log presuperscript 3 space x end cell equals 3 row cell log presuperscript 3 space x end cell equals cell scriptbase log space 3 cubed end scriptbase presuperscript 3 end cell row x equals cell 3 cubed end cell row x equals 27 end table   

Oleh karena itu, jawaban yang benar adalah E.

Roboguru

Jika , tentukanlah .

Pembahasan Soal:

Untuk meyelesaikan soal perlu dilakukan permisalan sebagai berikut.

Misal fraction numerator log invisible function application a over denominator b minus c end fraction equals fraction numerator log invisible function application b over denominator c minus a end fraction equals fraction numerator log invisible function application c over denominator a minus b end fraction equals k sehingga akan didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application a over denominator b minus c end fraction end cell equals k row cell log invisible function application a end cell equals cell k left parenthesis b minus c right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application b over denominator c minus a end fraction end cell equals k row cell log invisible function application b end cell equals cell k left parenthesis c minus a right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator log invisible function application c over denominator a minus b end fraction end cell equals k row cell log invisible function application c end cell equals cell k left parenthesis a minus b right parenthesis end cell end table

Ingat sifat pada bentuk logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a plus scriptbase log invisible function application c end scriptbase presuperscript a equals scriptbase log invisible function application c end scriptbase presuperscript a dan scriptbase log invisible function application b to the power of m end scriptbase presuperscript a equals m times scriptbase log invisible function application b end scriptbase presuperscript a maka

table attributes columnalign right center left columnspacing 0px end attributes row cell log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis end cell equals cell log invisible function application a to the power of a plus log invisible function application b to the power of b plus log invisible function application c to the power of c end cell row blank equals cell a times log invisible function application a plus b times log invisible function application b plus c times log invisible function application c end cell row blank equals cell a times k left parenthesis b minus c right parenthesis plus b times k left parenthesis c minus a right parenthesis plus c times k left parenthesis a minus b right parenthesis end cell row blank equals cell a times left parenthesis k b minus k c right parenthesis plus b times left parenthesis k c minus k a right parenthesis plus c times left parenthesis k a minus k b right parenthesis end cell row blank equals cell left parenthesis k a b minus k a c right parenthesis plus left parenthesis k b c minus k a b right parenthesis plus left parenthesis k a c minus k b c right parenthesis end cell row blank equals cell k a b minus k a c plus k b c minus k a b plus k a c minus k b c end cell row blank equals 0 end table

Didapatkan log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis equals 0, ingat sifat pada logaritma yaitu scriptbase log invisible function application b end scriptbase presuperscript a equals scriptbase log invisible function application c end scriptbase presuperscript a maka b equals c sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell log invisible function application open parentheses a to the power of a times b to the power of b times c to the power of c close parentheses end cell equals 0 row cell log invisible function application left parenthesis a to the power of a times b to the power of b times c to the power of c right parenthesis end cell equals cell log invisible function application 1 end cell row cell a to the power of a times b to the power of b times c to the power of c end cell equals 1 end table

Jadi, dapat disimpulkan bahwa hasil dari a to the power of a times b to the power of b times c to the power of c adalah 1.

Roboguru

Nilai x yang memenuhi persamaan: , adalah...

Pembahasan Soal:

Sebelum menyelesaikan soal di atas, ingatlah sifat bentuk logaritma berikut:

left parenthesis straight i right parenthesis space log subscript straight a open parentheses bc close parentheses equals log subscript straight a open parentheses straight b close parentheses plus log subscript straight a open parentheses straight c close parentheses left parenthesis ii right parenthesis space straight a to the power of log subscript straight a open parentheses straight b close parentheses end exponent equals straight b left parenthesis iii right parenthesis space straight m space log subscript straight a open parentheses straight b close parentheses equals log subscript straight a open parentheses straight b to the power of straight m close parentheses 

Berdasarkan sifat-sifat diatas, penyelesaian persamaan logaritma pada soal adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell log space straight x end cell equals cell 1 third space log space 8 plus log space 9 minus 1 third space log space 27 end cell row blank equals cell log open parentheses 8 to the power of 1 third end exponent close parentheses plus log open parentheses 3 squared close parentheses minus log open parentheses 27 to the power of 1 third end exponent close parentheses end cell row blank equals cell log open parentheses cube root of 8 close parentheses plus log open parentheses 3 squared close parentheses minus log open parentheses cube root of 27 close parentheses end cell row blank equals cell log open parentheses 2 close parentheses plus 2 space log open parentheses 3 close parentheses minus log open parentheses 3 close parentheses end cell row blank equals cell log open parentheses 2 close parentheses plus log open parentheses 3 close parentheses end cell row blank equals cell log open parentheses 2 cross times 3 close parentheses end cell row blank equals cell log open parentheses 6 close parentheses. end cell end table 

Didapat lah persamaan logaritma yang lebih sederhana yaitu table attributes columnalign right center left columnspacing 0px end attributes row cell log space straight x end cell equals cell log space 6 end cell end table. Maka nilai x yang memenuhi persamaan logaritma tersebut adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell log space straight x end cell equals cell log space 6 end cell row straight x equals cell 10 to the power of log space 6 end exponent end cell row straight x equals cell 6. end cell end table  
 

Dengan demikian, nilai x yang memenuhi persamaan: log space x equals 1 third space log space 8 plus log space 9 minus 1 third space log space 27 adalah x equals 6.

Roboguru

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